Posts tagged with: range

Hi Mike. Can you please explain Practise test 1, section 4, question 14? I figured out that the range was the largest change (but only by physically calculating the mean and median etc.)

Is there a better way to do this?

Also, I’d just like to say THANK YOU THANK YOU THANK YOU for this book it really is amazing!!

Well, you have to consider every answer choice, but there are a few mental shortcuts that I think are helpful.

The main thing to recognize right away is that the data is in order, so calculating the median is very easy. The middle value is 12, and if you remove that outlier 24, the median will STILL be 12. (It’s useful to remember that, in general, outliers have little to no effect on the median—that’s why you usually hear about median household income instead of mean household income, for example.) So I can cross of median without calculating it, and then I can cross off choice D as well.

The data being neatly sorted also helps one to think about the mean. Notice how nicely clustered the data is: how the 13s, 14s and 15s pretty nicely balance out the 11s, 10s, and 9s. That tells you that the mean might not be exactly the same as the median, but it should still be very close to 12. So without adding up all the values, I know the average of 20 of the values should be about 12, which means their sum should be about 240 (average times # of numbers equals sum). Adding the erroneous 24 value in there won’t change the average much: \dfrac{\approx 240+24}{21}\approx 12.57. (Note that while this estimate is not completely accurate, it doesn’t matter because you’re just looking for the biggest change, not the exact change.)

The range, on the other hand, will change a lot! The range with the 24 measure in there is 24-8 = 16. Take that 24 out and the range becomes 16 - 8 = 8. That’s by far the biggest change.

PS: Glad you like the Math Guide! 🙂

Could you please explain Test #8, calculator-allowed section, number 28?

Thanks!

You have to evaluate both range and standard deviation here. The former you can calculate. The latter you just evaluate visually; You will never have to actually calculate a standard deviation on the SAT.

Let’s talk about range first. To calculate range you just subtract the smallest number in the set from the biggest number. In this case, that means that r_1=88-56=32 and r_2=112-80=32. The numbers in the second set are obviously higher, but the range in both sets is the same: the highest number is 32 greater than the lowest number. Therefore, you really only need to look at choices A and D—the only choices that say r_1=r_2.

Like range, standard deviation is a measure of variability—the bigger the standard deviation, the more spread out the values in a set are from the mean. Visually, you can look at both dot plots in this question and see that the distributions are quite different. In the first plot, there are a couple outliers, but generally the pulse rates are pretty tightly concentrated around the mean of 72.  In the second plot, the dots are much more spread out—if you picked a dot randomly in the second plot, you’d be just as likely to land on an extreme data point like 80 or 112 as you would on the mean of 96. Therefore, you can conclude that the standard deviations of the two sets are not the same (it will be bigger in the second plot). That makes choice D the only option.